
using a variety of tools from geometric analysis and spectral theory, we show that for the geometric generators of \(\Theta _{V,\varepsilon ,\delta }\)  one can provide explicit upper bounds for the grading of any irreducible solution relative to the reducible one. This uses the interpretation via Atiyah-Patodi-Singer index theory [1]} of the relative grading in terms of the spectral flow of a family of operators, which can be bounded by making explicit certain quantities that arise in the classical proof of compactness of the moduli space of solutions to the Seiberg-Witten equations.

From this bound, one obtains restrictions on the monopole Floer homology of the generators of \(\Theta _{V,\varepsilon ,\delta }\) . Using this, one obtains upper bounds on certain numerical homology cobordism invariants arising in \(\mathrm {Pin}(2)\) -monopole Floer homology for all elements \(\Theta _{V,\varepsilon ,\delta }\) . One then concludes because for the manifolds \(Y_n\)  such invariants are explicitly computed, and tend to infinity.

